Math101 · Independent Ontario learning library
Math101Parallel Lines
Parallel lines keep a constant separation and, in the coordinate plane, have equal direction and equal slopes when nonvertical.
Distinct parallel lines lie in the same plane and never intersect. Nonvertical parallel lines have equal slopes.
Direction is the key
Slope measures a line’s direction. Lines $y=m_1x+b_1$ and $y=m_2x+b_2$ are parallel when $m_1=m_2$ and $b_1\ne b_2$.
If both slope and intercept match, the equations describe the same line rather than two distinct parallel lines.
Testing equations
Rewrite equations in slope-intercept form or calculate slope from standard form. The lines
and
both have slope $-2/3$, but different intercepts, so they are parallel.
Writing a parallel line
Check the new line’s slope and verify that the given point satisfies it.
Standard-form shortcut
Lines $Ax+By=C_1$ and $Ax+By=C_2$ share the same left-side coefficients and therefore the same direction. When $C_1\ne C_2$, they are parallel.
More generally, proportional $A$ and $B$ coefficients produce equal slopes as long as the equations are not equivalent in all coefficients.
Common mistakes
Using negative reciprocal slopes. That condition describes perpendicular lines.
Calling identical equations parallel. Coincident lines share every point.
Forcing a slope onto a vertical line. Compare vertical equations directly.
Checking only the $x$-coefficient in standard form. Direction depends on the coefficient pair.
Quick self-check
- Are the lines distinct?
- Do their slopes match, or are both vertical?
- Does the new line pass through the required point?
- In a system, should the result be no solution or infinitely many?
